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1.
Dongmei Xiao 《Journal of Dynamics and Differential Equations》2008,20(4):961-980
In this paper we consider a five-parameter family of planar vector fields where μ = (μ
1, μ
2, μ
3, μ
4, μ
5), which is a small parameter vector, and c(0) ≠ 0. The family X
μ
represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X
μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li
and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations
of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals,
1991). After making some rescalings, we obtain the truncated systems of X
μ
. For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system
is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied
for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at
most three limit cycles occur in the small neighborhood of the Hopf singularity.
Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday 相似文献
2.
The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due
to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations,
where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed
transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium,
found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled
this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼(μ−μ
c
)−1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ
c
. In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional
maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species
ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising
due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement
with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling
laws of one-dimensional discrete dynamical systems with saddle-node bifurcations. 相似文献
3.
We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary
conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods,
the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller’s conjecture
(Müller, Calc. Var. 1:169–204, 1993) concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of
the primary branch connections that occur in this problem. We give a reformulation of Müller’s conjecture and suggest two
further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that
characterizes (k,3k) bifurcations.
相似文献
4.
Maik Nowak 《Rheologica Acta》2001,40(4):366-372
The first normal stress differences N
1 of a highly dilute cationic surfactant solution are investigated in a cone-and-plate rheometer. In continuation of a previous
paper (Nowak 1998), where the buildup of a shear induced structure in such a solution was attained after a reduced deformation,
the N
1 turned out to be in proportion to the square of the shear rate γ˙ reduced by a critical value γ˙
c
in a first range above γ˙
c
. At higher shear rates the N
1 tend to lower values than predicted by this relation.
Relaxation experiments were performed in the same geometry to determine the characteristic time scales of the shear induced
state's decay. In the lower range above &γdot;
c
the stress decay is a monoexponential process, while a second time constant has to be introduced to describe the relaxation
in that range, where the N
1 deviate from the parabolic dependence of the reduced shear rate.
Received: 10 May 1999 Accepted: 15 November 2000 相似文献
5.
The flow bifurcation scenario and heat transfer characteristics in grooved channels, are investigated by direct numerical simulations of the mass, momentum and energy equations, using the spectral element methods for increasing Reynolds numbers in the laminar and transitional regimes. The Eulerian flow characteristics show a transition scenario of two Hopf bifurcations when the flow evolves from a laminar to a time-dependent periodic and then to a quasi-periodic flow. The first Hopf bifurcation occurs to a critical Reynolds number Rec1 that is significantly lower than the critical Reynolds number for a plane-channel flow. The periodic and quasi-periodic flows are characterized by fundamental frequencies ω1 and m· ω1+n·ω2, respectively, with m and n integers. Friction factor and pumping power evaluations demonstrate that these parameters are much higher than the plane channel values. The time-average mean Nusselt number remains mostly constant in the laminar regime and continuously increases in the transitional regime. The rate of increase of this Nusselt number is higher for a quasi-periodic than for a periodic flow regime. This higher rate originates because better flow mixing develops in quasi-periodic flow regimes. The flow bifurcation scenario occurring in grooved channels is similar to the Ruelle-Takens-Newhouse transition scenario of Eulerian chaos, observed in symmetric and asymmetric wavy channels. 相似文献
6.
Kenneth R. Meyer Patrick McSwiggen Xiaojie Hou 《Journal of Dynamics and Differential Equations》2010,22(3):367-380
The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search
for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ
1 as t → −∞ and u(t) → γ
2 as t → ∞ where γ
1, γ
2 are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that
f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions
we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied. 相似文献
7.
This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and
crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch.
TheN
x andN
y fringe patterns representing displacement field, and the ΔN
x/Δx and ΔN
γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per
fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN
x andN
γ fringe patterns.
The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China 相似文献
8.
The existence and characteristics of shock wave triple points are examined. The analysis utilizes a single flow plane for
the three shocks and is local to the triple point. It applies when the flow is unsteady, three-dimensional, and the upstream
flow is nonuniform. Under more restrictive conditions, a relation is also derived for the ratio of the curvature of the Mach
stem to that of the reflected shock. For given values of the ratio of specific heats, γ, and the upstream Mach number, M
1, a solution window is established. A parametric set of solutions is generated within the window for γ = 1, 1.4, and 5/3 and for 16 values of M
1 ranging from solution onset to M
1 = 6.A solution can be one of three types, these stem from the velocity tangency condition along the slip stream. Topics are
addressed such as solution multiplicity, shock wave and slip stream orientation, the nature of the reflected wave (weak, strong,
inverted, normal), the nature of the Mach stem (weak, strong, normal), and differences due to changes in γ and M
1. 相似文献
9.
The dynamic flow behavior of polyamide-6 (PA-6) and a nanocomposite (PNC) based on it was studied. The latter resin contained
2 wt% of organoclay. The two materials were blended in proportions of 0, 25, 50, 75, and 100 wt% PNC. The dynamic shear rheological
properties of well-dried specimens were measured under N2 at T=240 °C, frequency ω=0.1–100 rad/s, and strains γ=10 and 40%. At constant T, γ, and ω the time sweeps resulted in significant
increases of the shear moduli. The γ and ω scans showed a complex rheological behavior of all clay-containing specimens. At
γ=10% the linear viscoelasticity was observed for all compositions only at ω>1 rad/s, while at γ=40% only for 0 and 25 wt%
of PNC. However, the effect was moderate, namely decreasing G′ and G′′ (at ω=6.28 rad/s; γ=50%) by 15 and 7.5%, respectively.
For compositions containing >25 wt% PNC two types of non-linearity were detected. At ω≤ωc=1.4 ± 0.2 rad/s yield stress provided evidence of a 3-D structure. At ω > ωc, G′ and G′′ were sensitive to shear history – the effect was reversible. From the frequency scans at ω > ωc the zero-shear relative viscosity vs concentration plot was constructed. The initial slope gave the intrinsic viscosity from
which the aspect ratio of organoclay particles, p=287 ± 9 was calculated, in agreement with the value calculated from the
reduced permeability data, p=286.
Received: 24 May 2001 Accepted: 27 August 2001 相似文献
10.
Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting
an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies
of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems,
associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact
periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of
a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to
reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of
two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong
resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of
period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact
systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T
in, T
on and T
out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact
motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the
vibro-impact systems exhibit very complicated quasi-periodic impact motions.
The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of
Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang. 相似文献
11.
CH double-pulsed PLIF measurement in turbulent premixed flame 总被引:1,自引:0,他引:1
The flame displacement speeds in turbulent premixed flames have been measured directly by the CH double-pulsed planar laser-induced
fluorescence (PLIF). The CH double-pulsed PLIF systems consist of two independent conventional CH PLIF measurement systems
and laser beams from each laser system are led to same optical pass using the difference of polarization. The highly time-resolved
measurements are conducted in relatively high Reynolds number turbulent premixed flames on a swirl-stabilized combustor. Since
the time interval of the successive CH PLIF can be selected to any optimum value for the purpose intended, both of the large
scale dynamics and local displacement of the flame front can be discussed. By selecting an appropriate time interval (100–200 μs),
deformations of the flame front are captured clearly. Successive CH fluorescence images reveal the burning/generating process
of the unburned mixtures or the handgrip structures in burnt gas, which have been predicted by three-dimensional direct numerical
simulations of turbulent premixed flames. To evaluate the local flame displacement speed directly from the successive CH images,
a flame front identification scheme and a displacement vector evaluation scheme are developed. Direct measurements of flame
displacement speed are conducted by selecting a minute time interval (≈30 μs) for different Reynolds number (Re
λ = 63.1–115.0). Local flame displacement speeds coincide well for different Reynolds number cases. Furthermore, comparisons
of the mean flame displacement speed and the mean fluid velocity show that the convection in the turbulent flames will affect
the flame displacement speed for high Reynolds number flames. 相似文献
12.
A. Alper Ozalp 《Heat and Mass Transfer》2008,45(1):31-46
Variable fluid property continuity, Navier–Stokes and energy equations are solved for roughness induced forced convective
laminar-transitional flow in a micropipe. Influences of Reynolds number, heat flux and surface roughness, on the momentum-energy
transport mechanisms and second-law of thermodynamics, are investigated for the ranges of Re = 1–2,000, Q = 5–100 W/m2 and ε = 1–50 μm. Numerical investigations put forward that surface roughness accelerates transition with flatter velocity profiles
and increased intermittency values (γ); such that a high roughness of ε = 50 μm resulted in transitional character at Re
tra = 450 with γ = 0.136. Normalized friction coefficient (C
f*) values showed augmentation with Re, as the evaluated C
f* are 1.006, 1.028 and 1.088 for Re = 100, 500 and 1,500, respectively, at ε = 1 μm, the corresponding values rise to C
f* = 1.021, 1.116 and 1.350 at ε = 50 μm. Heat transfer rates are also recorded to rise with Re and ε; moreover the growing influence of ε on Nusselt number with Re is determined by the Nu
ε=50 μm/Nu
ε=1 μm ratios of 1.086, 1.168 and 1.259 at Re = 500, 1,000 and 1,500. Thermal volumetric entropy generation values decrease with Re and ε in heating; however the contrary is recorded for frictional volumetric entropy generation data, where the augmentations in are more considerable when compared with the decrease rates of 相似文献
13.
Turbulent wall pressure fluctuation measurements on a towed model at high Reynolds numbers 总被引:1,自引:1,他引:0
Turbulent wall pressure fluctuation measurements were made in water on a towed model of length 129.8 (m) and diameter 3.8
(cm) for steady speeds from 6.2 (m/s) to 15.5 (m/s). The drag on the model was measured with a strut mounted load cell which
provided estimates of the momentum thickness and friction velocity. Momentum thickness Reynolds numbers Re
θ varied from 4.8 × 105 to 1.1 × 106. The ratio of momentum thickness to viscous length scale is significantly greater than for flat plate cases at comparable
Re
θ. The effectiveness of inner and outer velocity and length scales for collapsing the pressure spectra are discussed. The wavenumber–frequency
spectra show a convective ridge at higher frequencies similar to flat plate boundary layers. At low frequencies, energy broad
in wavenumber extends outside the convective ridge and acoustic cone, with no characteristic wave speed. Wall pressure cross-spectral
levels scaled with similarity variables are shown to increase with increasing tow speed, and to follow decay constants consistent
with flat plate cases. The convection velocities also display features similar to flat plate cases. 相似文献
14.
Yieh-Hei Wan 《Journal of Dynamics and Differential Equations》1992,4(2):341-374
Consider a Hamiltonian system
with parameters, such that there exists an involution which reverses this Hamiltonian system. Let us assume the linear part L at =0 has only nonzero purely imaginary eigen-values ±ib1,..., ±ibn. In this paper, we classify the typical bifurcations of families of symmetric periodic solutions of this system at resonance if bi/bj=±1, ±2, or ±1/2 and the number of parameters needed is one or two. First, one puts the Hamiltonians into a convenient normal form. Next, applying a Lyapunov-Schmidt reduction and making further manipulations, one can geta reduced bifurcation equation which can possess certain symmetry. Finally, by using elementary methods from singularity theory or isotopy methods, one obtains the desired bifurcation diagrams. 相似文献
15.
Ji-Cheng Gu James R. Rice Andy L. Ruina Simon T. Tse 《Journal of the mechanics and physics of solids》1984,32(3):167-196
We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V0 We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V0 to change from stable to unstable with a decrease in the spring stiffness k below a critical value kcr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V0 is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations. 相似文献
16.
Hiroshi Watanabe Ming-Long Yao Kunihiro Osaki Toshiyuki Shikata Hirokazu Niwa Yotaro Morishima 《Rheologica Acta》1999,38(1):2-13
Nonlinear rheology was examined for concentrated suspensions of spherical silica particles (with radius of 40 nm) in viscous
media, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture and pure ethylene glycol. The particles were randomly and isotropically
dispersed in the media in the quiescent state, and their effective volume fraction φeff ranged from 0.36 to 0.59. For small strains, the particles exhibited linear relaxation of the Brownian stress σB due to their diffusion. For large step strains γ, the nonlinear relaxation modulus G(t,γ) exhibited strong damping and obeyed the time-strain separability. This damping was related to γ-insensitivity of strain-induced
anisotropy in the particle distribution that resulted in decreases of σB/γ. The damping became stronger for larger φeff. This φeff dependence was related to a hard-core volume effect, i.e., strain-induced collision of the particles that is enhanced for
larger φeff. Under steady/transient shear flow, the particles exhibited thinning and thickening at low and high γ˙, respectively. The thinning behavior was well described by a BKZ constitutive equation using the G(t,γ) data and attributable to decreases of a Brownian contribution, σB/γ˙. The thickening behavior, not described by this equation, was related to dynamic clustering of the particles and corresponding
enhancement of the hydrodynamic stress at high γ˙. In this thickening regime, the viscosity growth η+ after start-up of flow was scaled with a strain γ˙t. Specifically, critical strains γd and γs for the onset of thickening and achievement of the steadily thickened state were independent of γ˙ but decreased with increasing φeff. This φeff dependence was again related to the hard-core volume effect, flow-induced collision of the particles enhanced for larger
φeff.
Received: 26 June 1998 Accepted: 9 December 1998 相似文献
17.
Chien-Hsin Chen 《Heat and Mass Transfer》2006,42(9):853-860
Forced convection flow in a microchannel with constant wall temperature is studied, including viscous dissipation effect. The slip-flow regime is considered by incorporating both the velocity-slip and the temperature-jump conditions at the surface. The energy equation is solved for the developing temperature field using finite integral transform. To increase βv
Kn is to increase the slip velocity at the wall surface, and hence to decrease the friction factor. Effects of the parameters βv
Kn, β, and Br on the heat transfer results are illustrated and discussed in detail. For a fixed Br, the Nusselt number may be either higher or lower than those of the continuum regime, depending on the competition between the effects of βv
Kn and β. At a given βv
Kn the variation of local Nusselt number becomes more even when β becomes larger, accompanied by a shorter thermal entrance length. The fully developed Nusselt number decreases with increasing β irrelevant to βv
Kn. The increase in Nusselt number due to viscous heating is found to be more pronounced at small βv
Kn. 相似文献
18.
M. P. Escudier A. Abdel-Hameed M. W. Johnson C. J. Sutcliffe 《Experiments in fluids》1998,25(5-6):491-502
Experimental results are reported for the response of an initially turbulent boundary layer (Re
θ≈1700) to a favourable pressure gradient with a peak value of K≡(−υ/ρU
3
E
) dp/dx equal to 4.4×10-6. In the near-wall region of the boundary layer (y/δ<0.1) the turbulence intensity u′ scales roughly with the free-stream velocity U
E
until close to the location where K is a maximum whereas in the outer region u′ remains essentially frozen. Once the pressure gradient is relaxed, the turbulence level increases throughout the boundary
layer until K falls to zero when the near wall u′ levels show a significant decrease. The intermittency γ is the clearest indicator of a fundamental change in the turbulence
structure: once K exceeds 3×10-6, the value of γ in the immediate vicinity of the wall γ
s
falls rapidly from unity, reaches zero at the location where K again falls below 3×10-6 and then rises back to unity. Although γ is practically zero throughout the boundary layer in the vicinity of γ
s
=0, the turbulence level remains high. The explanation for what appears to be a contradiction is that the turbulent frequencies
are too low to induce turbulent mixing. The mean velocity profile changes shape abruptly where K exceeds 3×10-6. Values for the skin friction coefficient, based upon hot-film measurements, peak at the same location as K and fall to a minimum close to the location where K drops back to zero.
Received: 28 January 1998/Accepted: 8 April 1998 相似文献
19.
This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible
Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres.
The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams.
Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case
of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs
is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789.
This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another
new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated
by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re
1
, Re
2
) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation
theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states
are accompanied by certain unstable states.
Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002
Communicated by M.Y. Hussaini 相似文献
20.
High-speed tomographic PIV was used to investigate the coalescence of drops placed on a liquid/liquid interface; the coalescence
of a single drop and of a drop in the presence of an adjacent drop (side-by-side drops) was investigated. The viscosity ratio
between the drop and surrounding fluids was 0.14, the Ohnesorge number (Oh = μd/(ρdσD)1/2) was 0.011, and Bond numbers (Bo = (ρ
d
− ρ
s
)gD
2/σ) were 3.1–7.5. Evolving volumetric velocity fields of the full coalescence process allowed for quantification of the velocity
scales occurring over different time scales. For both single and side-by-side drops, the coalescence initiates with an off-axis
film rupture and film retraction speeds an order of magnitude larger than the collapse speed of the drop fluid. This is followed
by the formation and propagation of an outward surface wave along the coalescing interface with wavelength of approximately
2D. For side-by-side drops, the collapse of the first drop is asymmetric due to the presence of the second drop and associated
interface deformation. Overall, tomographic PIV provides insight into the flow physics and inherent three-dimensionalities
in the coalescence process that would not be achievable with flow visualization or planar PIV only. 相似文献